{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Линейная регрессия и шифрование обратной матрицей для защиты персональных данных"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Описание проекта**\n",
    "\n",
    "Вам нужно защитить данные клиентов страховой компании «Хоть потоп». Разработайте такой метод преобразования данных, чтобы по ним было сложно восстановить персональную информацию. Обоснуйте корректность его работы.\n",
    "\n",
    "Нужно защитить данные, чтобы при преобразовании качество моделей машинного обучения не ухудшилось. Подбирать наилучшую модель не требуется."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Инструкция по выполнению проекта**\n",
    "\n",
    "1. Загрузите и изучите данные.\n",
    "1. Ответьте на вопрос и обоснуйте решение. \n",
    " 1. Признаки умножают на обратимую матрицу. Изменится ли качество линейной регрессии? (Её можно обучить заново.)\n",
    " 1. Изменится. Приведите примеры матриц.\n",
    " 1. Не изменится. Укажите, как связаны параметры линейной регрессии в исходной задаче и в преобразованной.\n",
    "1. Предложите алгоритм преобразования данных для решения задачи. Обоснуйте, почему качество линейной регрессии не поменяется.\n",
    "1. Запрограммируйте этот алгоритм, применив матричные операции. Проверьте, что качество линейной регрессии из sklearn не отличается до и после преобразования. Примените метрику R2.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Описание данных**\n",
    "\n",
    "Набор данных находится в файле `/datasets/insurance.csv`.\n",
    "\n",
    "- ***Признаки***: пол, возраст и зарплата застрахованного, количество членов его семьи.\n",
    "- ***Целевой признак***: количество страховых выплат клиенту за последние 5 лет."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "toc": true
   },
   "source": [
    "<h1>Содержание<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Загрузка-данных\" data-toc-modified-id=\"Загрузка-данных-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Загрузка данных</a></span></li><li><span><a href=\"#Умножение-матриц\" data-toc-modified-id=\"Умножение-матриц-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Умножение матриц</a></span></li><li><span><a href=\"#Алгоритм-преобразования\" data-toc-modified-id=\"Алгоритм-преобразования-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>Алгоритм преобразования</a></span></li><li><span><a href=\"#Проверка-алгоритма\" data-toc-modified-id=\"Проверка-алгоритма-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>Проверка алгоритма</a></span></li><li><span><a href=\"#Выводы-проекта\" data-toc-modified-id=\"Выводы-проекта-5\"><span class=\"toc-item-num\">5&nbsp;&nbsp;</span>Выводы проекта</a></span></li><li><span><a href=\"#Чек-лист-проверки\" data-toc-modified-id=\"Чек-лист-проверки-6\"><span class=\"toc-item-num\">6&nbsp;&nbsp;</span>Чек-лист проверки</a></span></li></ul></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Загрузка данных"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Базовые библиотеки\n",
    "import pandas as pd # Датафреймы\n",
    "import numpy as np # Математика для массивов\n",
    "\n",
    "# Автоматизация раздельного декодирования признаков\n",
    "from sklearn.compose import(\n",
    "    make_column_selector, \n",
    "    make_column_transformer, \n",
    "    ColumnTransformer\n",
    ")\n",
    "\n",
    "# Обработка данных для машинного обучения\n",
    "# Стандартизация данных\n",
    "import re\n",
    "from sklearn.preprocessing import(\n",
    "    StandardScaler\n",
    ")\n",
    "\n",
    "from sklearn.model_selection import(\n",
    "    train_test_split, # Разделение выборок с целевыми и нецелевыми признаками на обучающую и тестовую\n",
    ") \n",
    "\n",
    "# Метрики (Показатели качества моделей)\n",
    "from sklearn.metrics import(\n",
    "    r2_score, # R^2, Коэффициент детерминации (универсальная в %, чувствительная к выбросам, может быть отрицательной и возвращать NaN)\n",
    ")\n",
    "\n",
    "# Визуализация графиков\n",
    "import seaborn as sns\n",
    "import matplotlib\n",
    "%matplotlib inline\n",
    "from matplotlib import pyplot as plt\n",
    "from matplotlib import rcParams, rcParamsDefault\n",
    "from pandas.plotting import scatter_matrix\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Отображение всех столбцов таблицы\n",
    "pd.set_option('display.max_columns', None)\n",
    "# Обязательно для нормального отображения графиков plt\n",
    "rcParams['figure.figsize'] = 10, 6\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "# Дополнительно и не обязательно для декорирования графиков\n",
    "factor = .8\n",
    "default_dpi = rcParamsDefault['figure.dpi']\n",
    "rcParams['figure.dpi'] = default_dpi * factor\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Чтение данных из файла\n",
    "try:\n",
    "    data = pd.read_csv('/datasets/insurance.csv') \n",
    "except:\n",
    "    data = pd.read_csv('datasets/insurance.csv') \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "RangeIndex: 5000 entries, 0 to 4999\n",
      "Data columns (total 5 columns):\n",
      " #   Column             Non-Null Count  Dtype  \n",
      "---  ------             --------------  -----  \n",
      " 0   Пол                5000 non-null   int64  \n",
      " 1   Возраст            5000 non-null   float64\n",
      " 2   Зарплата           5000 non-null   float64\n",
      " 3   Члены семьи        5000 non-null   int64  \n",
      " 4   Страховые выплаты  5000 non-null   int64  \n",
      "dtypes: float64(2), int64(3)\n",
      "memory usage: 195.4 KB\n",
      "None\n"
     ]
    },
    {
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       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Пол</th>\n",
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       "      <th>Члены семьи</th>\n",
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       "   Пол  Возраст  Зарплата  Члены семьи  Страховые выплаты\n",
       "0    1     41.0   49600.0            1                  0\n",
       "1    0     46.0   38000.0            1                  1\n",
       "2    0     29.0   21000.0            0                  0\n",
       "3    0     21.0   41700.0            2                  0\n",
       "4    1     28.0   26100.0            0                  0"
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     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Изучение датафрейма\n",
    "print(data.info()) \n",
    "data.head() \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
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   "outputs": [
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       "      <th>Пол</th>\n",
       "      <th>Возраст</th>\n",
       "      <th>Зарплата</th>\n",
       "      <th>Члены семьи</th>\n",
       "      <th>Страховые выплаты</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>5000.000000</td>\n",
       "      <td>5000.000000</td>\n",
       "      <td>5000.000000</td>\n",
       "      <td>5000.000000</td>\n",
       "      <td>5000.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>0.499000</td>\n",
       "      <td>30.952800</td>\n",
       "      <td>39916.360000</td>\n",
       "      <td>1.194200</td>\n",
       "      <td>0.148000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>0.500049</td>\n",
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       "      <td>1.091387</td>\n",
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       "      <td>0.000000</td>\n",
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       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
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       "    <tr>\n",
       "      <th>25%</th>\n",
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       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
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       "    <tr>\n",
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       "      <th>75%</th>\n",
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       "    <tr>\n",
       "      <th>max</th>\n",
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       "               Пол      Возраст      Зарплата  Члены семьи  Страховые выплаты\n",
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       "min       0.000000    18.000000   5300.000000     0.000000           0.000000\n",
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       "75%       1.000000    37.000000  46600.000000     2.000000           0.000000\n",
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       "<Figure size 800x480 with 6 Axes>"
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     "metadata": {
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     "output_type": "display_data"
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   ],
   "source": [
    "# Изучение распределений данных\n",
    "data.hist();\n",
    "data.describe() \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Страховые выплаты\n",
       "0    4436\n",
       "1     423\n",
       "2     115\n",
       "3      18\n",
       "4       7\n",
       "5       1\n",
       "Name: Страховые выплаты, dtype: int64"
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     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "# Количество значений \n",
    "# разных страховых выплат\n",
    "#data['Страховые выплаты'].unique() \n",
    "data.groupby(by='Страховые выплаты', axis=0)['Страховые выплаты'].count() \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
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       "                        Пол    Возраст      Зарплата  Члены семьи\n",
       "Страховые выплаты                                                \n",
       "0                  0.499324  28.970694  39924.752029     1.205816\n",
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     "execution_count": 7,
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   "source": [
    "# Средние значения показателей\n",
    "# для разного количества страховых выплат\n",
    "data.groupby(by='Страховые выплаты', axis=0).mean() \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
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       "                        Пол   Возраст  Зарплата  Члены семьи  \\\n",
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       "Зарплата           0.014910 -0.019093  1.000000    -0.030296   \n",
       "Члены семьи       -0.008991 -0.006692 -0.030296     1.000000   \n",
       "Страховые выплаты  0.010140  0.651030 -0.014963    -0.036290   \n",
       "\n",
       "                   Страховые выплаты  \n",
       "Пол                         0.010140  \n",
       "Возраст                     0.651030  \n",
       "Зарплата                   -0.014963  \n",
       "Члены семьи                -0.036290  \n",
       "Страховые выплаты           1.000000  "
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     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Корреляция значения показателей\n",
    "data.corr() \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Итоги анализа загруженных данных**\n",
    "\n",
    "1. Датафрейм содержит 5000 объектов и 5 числовых признаков. \n",
    "1. В данных нет пропусков. \n",
    "1. Целевой признак содержит 6 категорий от `0` до `5`. Вероятнее всего, категории обозначают количество страховых выплат. По мере увеличения значения категории уменьшается количество объектов в ней. Так, в категории `0` содержится 4436 объектов, в категории `1` только 423 объекта, а в категории `5` всего 1 объект. \n",
    "1. По мере увеличения количества страховых выплат увеличивается возраст застрахован лица. Эти два показателя имеют сильную положительную корреляцию. При этом, количество страховых выплат почти не зависит от пола и других имеющихся признаков застрахованного лица. \n",
    "1. Признаки имеют значительное различие в разбросе числовых значений. Для эффективного использования в обучении их следует стандартизировать. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Умножение матриц"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "В этом задании вы можете записывать формулы в *Jupyter Notebook.*\n",
    "\n",
    "Чтобы записать формулу внутри текста, окружите её символами доллара \\\\$; если снаружи —  двойными символами \\\\$\\\\$. Эти формулы записываются на языке вёрстки *LaTeX.* \n",
    "\n",
    "Для примера мы записали формулы линейной регрессии. Можете их скопировать и отредактировать, чтобы решить задачу.\n",
    "\n",
    "Работать в *LaTeX* необязательно."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Обозначения:\n",
    "\n",
    "- $X$ — матрица признаков (нулевой столбец состоит из единиц)\n",
    "\n",
    "- $y$ — вектор целевого признака\n",
    "\n",
    "- $P$ — матрица, на которую умножаются признаки\n",
    "\n",
    "- $w$ — вектор весов линейной регрессии (нулевой элемент равен сдвигу)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Предсказания:\n",
    "\n",
    "$$\n",
    "a = Xw\n",
    "$$\n",
    "\n",
    "Задача обучения:\n",
    "\n",
    "$$\n",
    "w = \\arg\\min_w MSE(Xw, y)\n",
    "$$\n",
    "\n",
    "Формула обучения:\n",
    "\n",
    "$$\n",
    "w = (X^T X)^{-1} X^T y\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Вопрос:**\n",
    "\n",
    "Признаки умножают на обратимую матрицу. Изменится ли качество линейной регрессии? (Её можно обучить заново.)\n",
    "\n",
    "**Ответ:**\n",
    "\n",
    "Не изменится.\n",
    "\n",
    "**Обоснование:**\n",
    "\n",
    "Если использовать формулу линейной регрессии (https://academy.yandex.ru/handbook/ml/article/linear-models):\n",
    "\n",
    "$$\n",
    "a = Xw\n",
    "$$\n",
    "$$\n",
    "w = (X^T X)^{-1} X^T y\n",
    "$$\n",
    "\n",
    "то, при умножжении признаков на обратимую матрицу, формула примет вид:\n",
    "\n",
    "$$\n",
    "a' = X'w' = (XP)w'\n",
    "$$\n",
    "$$\n",
    "w' = ((XP)^T (XP))^{-1} (XP)^T y\n",
    "$$\n",
    "\n",
    "где $P$ - случайная обратимая матрица. Преобразуем форумлу:\n",
    "\n",
    "$$\n",
    "w' = (X'^T X')^{-1} X'^T y\n",
    "$$\n",
    "$$\n",
    "w' = ((X P)^T (X P))^{-1} (X P)^T y\n",
    "$$\n",
    "$$\n",
    "w' = (P^T X^T X P)^{-1} P^T X^T y\n",
    "$$\n",
    "$$\n",
    "w' = P^{-1} (X^T X)^{-1} (P^T)^{-1} P^T X^T y\n",
    "$$\n",
    "$$\n",
    "w' = P^{-1} (X^T X)^{-1} E X^T y\n",
    "$$\n",
    "$$\n",
    "w' = P^{-1} (X^T X)^{-1} X^T y\n",
    "$$\n",
    "$$\n",
    "w' = P^{-1} w\n",
    "$$\n",
    "\n",
    "Т.к. $w' \\not= w$, то и $a' \\not= a$, однако, если $w'$ можно выразить через $P$ и $w$, то и $a'$ можно выразить через $X$, $P$ и $w$:\n",
    "\n",
    "$$\n",
    "a' = X' w'\n",
    "$$\n",
    "$$\n",
    "a' = (X P) P^{-1} w\n",
    "$$\n",
    "$$\n",
    "a' = X P P^{-1} w\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Алгоритм преобразования"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Алгоритм**\n",
    "\n",
    "1. При вызове метода `fit()` пользовательского класса `LinearRegression()` внутри метода:\n",
    " 1. Создать внутриклассовую переменную `self.P` со случайной обратимой матрицей.\n",
    " 1. Проверить случайную матрицу на обратимость. Для этого проверить определитель матрицы на равенство нулю. В случае равенства нулю, создать матрицу заново. \n",
    " 1. Создать переменную `X` со скалярным произведением обучающей выборки `train_features`, содержащей все аттрибуты, кроме целевого, на `self.P`.\n",
    " 1. Добавить единичный столбец в `X`.\n",
    " 1. Создать переменную `y`, в которую передать данные целевого признака обучающей выборки `train_target`.\n",
    " 1. Создать переменную `w` и записать в нее результаты выполнения формулы $w = (X^T X)^{-1} X^T y$ с использованием одноименных переменных метода `fit()`.\n",
    " 1. Создать внутриклассовые переменные `self.w` и `self.w0`, в которые передать все значения из переменной `w`, кроме нулевого, и нулевое, соответственно.\n",
    "1. При вызове метода `predict()` пользовательского класса `LinearRegression()`:\n",
    " 1. Создать переменную `X` со скалярным произведением обучающей выборки `test_features`, содержащей все аттрибуты, кроме целевого, на `self.P`.\n",
    " 1. Вернуть в качестве предсказания, результаты скалярного произведения переменной `X` на внутриклассовую переменную `self.w`, сложенного с внутриклассовой переменной `self.w0`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Обоснование**\n",
    "\n",
    "На основании выводов, полученных в пункте 2 данного проекта предполагается, что умножение треннировочной выборки с нецелевыми аттрибутами и аналогичной тестовой выборки на случайную обратимую матрицу даст такой же результат, что и без умножения на подобную матрицу."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Проверка алгоритма"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "# Создание класса \"LinearRegression\" и его методов\n",
    "# для предсказания результатов методом линейной регрессии\n",
    "class LinearRegression:\n",
    "    \n",
    "    # Метод, включающий использование \n",
    "    # случайной обратимой матрицы для защиты данных\n",
    "    def keep(self, keeping):\n",
    "        if keeping == True:\n",
    "            self.keep = True\n",
    "        else:\n",
    "            self.keep = False\n",
    "    \n",
    "    # Метод обучения\n",
    "    def fit(self, train_features, train_target):\n",
    "        n = 0\n",
    "        while n == 0:\n",
    "            self.P = np.random.normal(size=(len(train_features.columns), len(train_features.columns)))\n",
    "            n = np.linalg.det(self.P) \n",
    "        if self.keep == True:\n",
    "            X = train_features @ self.P\n",
    "            X = np.concatenate((np.ones((X.shape[0], 1)), X), axis=1)\n",
    "        else:\n",
    "            X = np.concatenate((np.ones((train_features.shape[0], 1)), train_features), axis=1)\n",
    "        y = train_target\n",
    "        w = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(y)\n",
    "        self.w = w[1:]\n",
    "        self.w0 = w[0]\n",
    "    \n",
    "    # Метод предсказания\n",
    "    def predict(self, test_features):\n",
    "        if self.keep == True:\n",
    "            X = test_features @ self.P\n",
    "        else:\n",
    "            X = test_features\n",
    "        return X.dot(self.w) + self.w0\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Подготовка данных для использования в модели\n",
    "\n",
    "# Разделение датафрейма на выбоки с целевыми и не целевыми аттрибутами\n",
    "features = data.drop('Страховые выплаты', axis=1)\n",
    "target = data['Страховые выплаты']\n",
    "\n",
    "# Разделение выборок на обучающие и треннировочные\n",
    "features_train,  features_test, target_train, target_test = train_test_split(\n",
    "    features, \n",
    "    target, \n",
    "    test_size=.25, \n",
    "    random_state=12345\n",
    ")\n",
    "\n",
    "# Стандартизация данных \n",
    "scaler = StandardScaler()\n",
    "scaler.fit(features_train)\n",
    "features_train = scaler.transform(features_train)\n",
    "features_test = scaler.transform(features_test)\n"
   ]
  },
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   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
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      "R2: 0.42494550286668\n"
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   "source": [
    "# Реализаця модели без защиты данных\n",
    "model = LinearRegression()\n",
    "model.keep(False)\n",
    "model.fit(features, target)\n",
    "predictions = model.predict(features)\n",
    "\n",
    "pd.DataFrame(\n",
    "    data={\n",
    "        'predictions': predictions.values,\n",
    "        'target': target.values,\n",
    "    }\n",
    ").plot(\n",
    "    kind='hist',  \n",
    "    alpha=.5, \n",
    "    rot=20,\n",
    "    bins=20\n",
    ")\n",
    "\n",
    "r2 = []\n",
    "r2.append(r2_score(target, predictions))\n",
    "print('R2:', r2[0])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R2: 0.4249454692806999\n"
     ]
    },
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      "text/plain": [
       "<Figure size 800x480 with 1 Axes>"
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     },
     "metadata": {
      "needs_background": "light"
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     "output_type": "display_data"
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   ],
   "source": [
    "# Реализаця модели с защитой данных\n",
    "model = LinearRegression()\n",
    "model.keep(True)\n",
    "model.fit(features, target)\n",
    "predictions = model.predict(features)\n",
    "\n",
    "pd.DataFrame(\n",
    "    data={\n",
    "        'predictions': predictions.values,\n",
    "        'target': target.values,\n",
    "    }\n",
    ").plot(\n",
    "    kind='hist',  \n",
    "    alpha=.5, \n",
    "    rot=20,\n",
    "    bins=20\n",
    ")\n",
    "\n",
    "r2.append(r2_score(target, predictions))\n",
    "print('R2:', r2[1])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Показатель R2 моделей:\n",
      "с защищенными данными  : 0.4249454692806999\n",
      "с незащищенными данными: 0.42494550286668\n",
      "\n",
      "Разница в показателях  : -3.358598010816394e-08\n"
     ]
    }
   ],
   "source": [
    "# Разница показателя \"R2\" между моделями \n",
    "# с защищенными и незащищенными данными\n",
    "print('Показатель R2 моделей:')\n",
    "print('с защищенными данными  :', r2[1])\n",
    "print('с незащищенными данными:', r2[0])\n",
    "print()\n",
    "print('Разница в показателях  :', r2[1] - r2[0])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Вывод из проверки алгоритма**\n",
    "\n",
    "Разница показателя `R2` между моделями с защищенными и не защищенными данными стремиться к нулю. Можно сделать вывод о том, что защиту данных с помощью случайной обратимой матрицы допустимо использовать. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Выводы проекта"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Цель проекта достигнута: обоснован, создан и проверен класс `LinearRegression()` для предсказания количества страховых случаев с защитой использованных в классе данных путем их умножения на случайную обратимую матрицу. В рамках проекта решены следующие задачи:\n",
    "1. Загружены и проанализированы данные.\n",
    "1. Найден и обоснован ответ на вопрос о зависимости `w` и `w'` с использованием случайной обратимой матрицы.\n",
    "1. Предложен и обоснован алгоритм решения данных, используемых в классе `LinearRegression()`, путем их умножения на случайную обратимую матрицу.\n",
    "1. Эксперементально доказана возможнгость использования случайной обратимой матрицы для защиты данных в классе `LinearRegression()`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Чек-лист проверки"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Поставьте 'x' в выполненных пунктах. Далее нажмите Shift+Enter."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- [x]  Jupyter Notebook открыт\n",
    "- [x]  Весь код выполняется без ошибок\n",
    "- [x]  Ячейки с кодом расположены в порядке исполнения\n",
    "- [x]  Выполнен шаг 1: данные загружены\n",
    "- [x]  Выполнен шаг 2: получен ответ на вопрос об умножении матриц\n",
    "    - [x]  Указан правильный вариант ответа\n",
    "    - [x]  Вариант обоснован\n",
    "- [x]  Выполнен шаг 3: предложен алгоритм преобразования\n",
    "    - [x]  Алгоритм описан\n",
    "    - [x]  Алгоритм обоснован\n",
    "- [x]  Выполнен шаг 4: алгоритм проверен\n",
    "    - [x]  Алгоритм реализован\n",
    "    - [x]  Проведено сравнение качества моделей до и после преобразования"
   ]
  }
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    "duration": 76,
    "start_time": "2023-07-30T03:13:25.418Z"
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